Young children are very familiar with differences in amounts. For one thing, they see children of many ages: small like a baby, bigger like a younger sister, and still bigger like an older brother. They are also familiar with ideas like adding on, as in the case with the child who asks for more candies, and then more again. Although crude, these imprecise ideas form the basis for regularly growing patterns; that is, patterns that grow by specific amounts.

Jean Piaget was probably the first to investigate seriation, the type of growing pattern which involves arranging objects in order of magnitude, that is, in a series. Here is four-year-old Maya trying to order Cuisenaire rods of different lengths.

During this interaction, I asked Maya to “make it go bigger and bigger each time.” This requires placing the rods so that each subsequent rod is longer than the one previously placed, and shorter than the one that will come next. Each rod (except the first and the last) is at the same time longer than the one preceding it and shorter than the one next in the series. Piaget found that young children struggle with this coordination of relations. At the outset, Maya carefully lines up the first four sticks: white, red, purple, and then yellow. After this, she notices that she has forgotten the green stick, and then quickly fixes the sequence by putting the stick in its appropriate position to the right of the next-smaller but to the left of the next-larger stick. (This correction is similar to the kind of analysis Ben did to determine a missing object’s place in a pattern.) Maya then is faced with the task of placing the remaining four sticks into their proper positions in the sequence. She reaches out, touches the black stick very quickly, waves her hand briefly above the orange, touches the blue but puts it down quickly, and then selects the black one, which is the correct choice. How did she know that black was next? One possibility is that she realized that the smallest of the sticks not chosen would have to be the largest in the sequence of those already chosen. The black stick was both longer than the yellow as well as shorter than the brown, blue, and orange sticks. Even an apparently simple task like placing sticks in order of size requires some careful thinking.

To recapitulate: young children are fascinated by differences in size, length, number, and much else. These differences, if placed in order, can be considered a pattern. Constructing and understanding growing patterns like these requires a good deal of careful thought.